Question

Please can someone help me with this 2 part question

Answer 1

Step 1:

Write the matrix:

`\begin{gathered} X\text{ = }\begin{bmatrix}{6} & {a} \\ {-2} & {-5}\end{bmatrix} \\ I\text{ = }\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \end{gathered}`

Step 2:

Find the inverse of X.

`X^(-1)\text{ = }(Adjoint)/(Determinant)`

Step 3:

Find the determinant

`\begin{gathered} Determinant\text{ of X = }\begin{bmatrix}{6} & {a} \\ {-2} & {-5}\end{bmatrix} \\ Co-factor\text{ of X = }\begin{bmatrix}{-5} & {2} \\ {-a} & {6}\end{bmatrix} \\ Adjoint\text{ is the transpose of the co-factor of X} \\ Adj\text{oint = }\begin{bmatrix}{-5} & {-a} \\ {2} & {6}\end{bmatrix} \\ Deterninant\text{ = 2a - 30} \end{gathered}`

Step 4:

`\begin{gathered} X\text{ - }X^(-1)\text{ = I} \\ \\ \begin{bmatrix}{6} & {a} \\ {-2} & {-5}\end{bmatrix}\text{ - }\frac{\begin{bmatrix}{-5} & {-a} \\ {2} & {6}\end{bmatrix}}{2a\text{ - 30}}\text{ = }\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \\ \begin{bmatrix}{6} & {a} \\ {-2} & {-5}\end{bmatrix}\text{ - }\begin{bmatrix}{(-5)/(2a-30)} & {(-a)/(2a-30)} \\ {(2)/(2a-30)} & {(6)/(2a-30)}\end{bmatrix}\text{= }\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \\ \begin{bmatrix}{6\text{ + }(5)/(2a-30)} & {a\text{ + }\frac{a}{2a\text{ - 30}}} \\ {-2\text{ -}\frac{2}{2a\text{ - 30}}} & {-5-\text{ }\frac{6}{2a\text{ - 30}}}\end{bmatrix}\text{ = }\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \end{gathered}`

Step 5;

Write equations

`\begin{gathered} \text{ 6 + }\frac{5}{2a\text{ - 30 }}\text{ = 1} \\ \frac{6(2a\text{ - 30\rparen + 5}}{2a\text{ - 30}}\text{ = 1} \\ \frac{6(2a\text{ -30\rparen + 5}}{2a\text{ - 30}}\text{ = 1} \\ 12a\text{ - 180 + 5 = 2a - 30} \\ 12a\text{ - 2a = 175 - 30} \\ 10a\text{ = 145} \\ \text{a = }(145)/(10) \\ \text{a = }(29)/(2) \end{gathered}`

Final answer

`\begin{gathered} i)\text{ }X^(-1)\text{ = }\frac{1}{2a\text{ - 30}}\begin{bmatrix}{-5} & {-a} \\ {2} & {6}\end{bmatrix} \\ ii)\text{ }a\text{ = }(29)/(2)\text{ or 14.5} \end{gathered}`